If one root of the equation `(x-1)(7-x)=m` is three times the other, then `m` is equal to

Options

A. `-5`
B. `0`
C. `2`
D. `5`

Explanation

To solve the equation \((x-1)(7-x) = m\) given that one root is three times the other, we can follow these steps:

### Step 1: Expand the equation
We start with the equation:
\[ (x-1)(7-x) = m \]
Expanding the left side:
\[ x \cdot 7 - x^2 - 7 + x = m \]
This simplifies to:
\[ -x^2 + 8x - 7 = m \]
Rearranging gives us:
\[ -x^2 + 8x + (m - 7) = 0 \]
Multiplying through by -1:
\[ x^2 - 8x + (7 - m) = 0 \]

### Step 2: Let the roots be \(\alpha\) and \(3\alpha\)
Let the roots of the quadratic equation be \(\alpha\) and \(3\alpha\).

### Step 3: Use Vieta's formulas
According to Vieta's formulas, the sum of the roots \(\alpha + 3\alpha\) is equal to the coefficient of \(x\) with a negative sign:
\[ \alpha + 3\alpha = 8 \implies 4\alpha = 8 \implies \alpha = 2 \]

### Step 4: Find the product of the roots
The product of the roots \(\alpha \cdot 3\alpha\) is given by:
\[ \alpha \cdot 3\alpha = 3\alpha^2 \]
According to Vieta's formulas, this product is equal to \(7 - m\):
\[ 3\alpha^2 = 7 - m \]
Substituting \(\alpha = 2\):
\[ 3(2^2) = 7 - m \implies 3 \cdot 4 = 7 - m \implies 12 = 7 - m \]

### Step 5: Solve for \(m\)
Rearranging the equation gives:
\[ m = 7 - 12 \implies m = -5 \]

### Conclusion
Thus, the value of \(m\) is:
\[ \boxed{-5} \]